Title: The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
Author: Jim Pitman and Marc Yor
Pub: Annals of Probability 25, pages 855-900 (1997)
Date: August 1995
Abstract:
The two-parameter Poisson-Dirichlet distribution, denoted \PD$(\alpha,\theta)$,
is a distribution on the set of decreasing positive sequences with sum 1.
The usual Poisson-Dirichlet distribution with a single parameter $\theta$,
introduced by Kingman, is \PD$(0,\theta)$. Known properties of \PD$(0,\theta)$,
including the Markov chain description due to Vershik-Shmidt-Ignatov,
are generalized to the two-parameter case. The size-biased random permutation
of \PD$(\alpha,\theta)$ is a simple residual allocation model proposed by Engen
in the context of species diversity, and rediscovered by Perman and the authors
in the study of excursions of Brownian motion and Bessel processes.
For $0 < \alpha < 1$, \PD$(\alpha,0)$ is the asymptotic distribution of ranked
lengths of excursions of a Markov chain away from a state whose recurrence time
distribution is in the domain of attraction of a stable law of index $\alpha$.
Formulae in this case trace back to work of Darling, Lamperti and Wendel
in the 1950's and 60's. The distribution of ranked lengths of excursions of a
one-dimensional Brownian motion is \PD$(1/2,0)$, and the corresponding
distribution for Brownian bridge is \PD$(1/2,1/2)$. The \PD$(\alpha,0)$ and
\PD$(\alpha,\alpha)$ distributions admit a similar interpretation in terms of
the ranked lengths of excursions of a semi-stable Markov process whose zero set
is the range of a stable subordinator of index $\alpha$.